Maybe he saw Beckett spending a lot of time with another man without knowing it was for a case. Castle is a bit of a jealous type, and he might overreact to normal interactions.
There might be changes in Beckett's behavior that Castle misinterpreted. She could be stressed about a case and acting a bit differently, but Castle took it as signs of her being unfaithful. For instance, she might be more secretive about her work, which Castle wrongly thought was to hide an affair.
It could be that some misunderstandings were created on purpose by someone who wants to drive a wedge between them. For example, false rumors or doctored photos. Castle, without proper investigation, immediately jumped to the conclusion that Beckett was cheating.
It could be that Castle is overprotective. He loves Alexis so much that even the slightest sign of Beckett not being on the same page as Alexis in Castle's eyes makes him think Beckett hurt her. Maybe Beckett gave Alexis some career advice that Castle thought was too pushy.
Maybe in the fanfic, Beckett got a new job opportunity in a far - away place. This could make Castle think she's leaving him.
She is probably angry and frustrated. She would demand Castle to explain why he thinks so and would be hurt that he doesn't trust her.
He might be in denial at first. He could keep trying to convince himself that there's no way Beckett would leave him.
One possible reason could be that they are on a secret mission. Maybe they are trying to protect a key witness or important evidence from a powerful and dangerous criminal organization. In the world of fanfic, this could lead to them having to disappear to keep their whereabouts unknown.
Maybe he's afraid of losing her because she's his partner and they've been through so much together. Their relationship is very important to him.
In the fanfic, it could be that there are signs that Beckett is hiding something. Perhaps he has unexplained absences. He might say he's at work but Alexis can't reach him during those times. Also, he could be overly protective of his phone or personal space, which makes Alexis suspect that he's cheating.
Jealousy could also play a part. Castle is often surrounded by beautiful and adoring fans as a famous writer. Beckett might start to feel insecure, especially if she sees some of these fans getting a bit too close to Castle. On the flip side, Beckett's dedication to her work and her male partners in the police force could make Castle jealous, ultimately leading to their break - up.
Perhaps there have been some rumors or whispers in their fictional world that have reached Alexis' ears. Even if they are baseless, they can plant the seed of doubt in Alexis' mind, making her think that Kate is cheating.
Maybe they stumble upon a government conspiracy. The government wants to keep something hidden and they are seen as a threat, so they are made to disappear. It could also be related to a scientific experiment gone wrong. They are in the wrong place at the wrong time and get caught in an energy field that transports them to an unknown location, causing them to disappear from their normal world.